数域上的椭圆曲线与整数分解

This project is about elliptic curve over number field and integer factoring. Up to now, we have proved, for a given rational integer D with two distinct odd prime factors, we could construct a family of elliptic curves which have conjectural rank one under the parity conjecture, and we can factor integer D using the x-coordinate of Q, which is the generator of the free part of Mordell-Weil group. In the following, we will continue to study the complexity of this factoring method and the case which is not rank one. Otherwise, we will also discuss the similar factoring problem over quadratic number fields.

本项目旨在研究数域上的椭圆曲线与代数整数分解的问题。目前我们已经证明，对于给定的含两个不同奇素因子的有理整数D，我们可以构造出一族在奇偶性猜想下秩为1的椭圆曲线，并且可以利用这些曲线的莫代尔-威伊自由部分的生成元Q的x-坐标实现整数D的分解，作为后续工作，我们将继续研究这一分解方法的复杂度问题及秩不为1的情形。再者，我们也将初步探讨二次数域上的整数分解问题。